Question: Let $h$ be a twice differentiable function, and let $h(6)=7$, $h'(6)=0$, and $h''(6)=0$. What occurs in the graph of $h$ at the point $(6,7)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(6,7)$ is a minimum point. (Choice B) B $(6,7)$ is a maximum point. (Choice C) C There's not enough information to tell.
Answer: Since $h'(6)=0$, we know that $x=6$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $h$ at this point according to these three cases: If $h''(6)>0$, the graph of $h$ has a minimum point at $x=6$. If $h''(6)<0$, the graph of $h$ has a maximum point at $x=6$. If $h''(6)=0$, the test is inconclusive. [Why is this so?] We are given that $h''(6)=0$. The test is inconclusive. There's not enough information to tell whether $(6,7)$ is a minimum point, a maximum point, or neither.